MAT 1332: CALCULUS FOR LIFE SCIENCESJING LIContents1.Review: Functions of several variables I: Introduction11.1.Functions of two or more independent variables11.2.The level set11.3.Limits and continuity12.Function of several variables II: Partial derivatives12.1.Definition12.2.Geometric interpretation of partial derivatives32.3.Linear approximation71.Review: Functions of several variables I: Introduction1.1.Functions of two or more independent variables.•Domain•Range1.2.The level set.1.3.Limits and continuity.2.Function of several variables II: Partial derivatives2.1.Definition.Suppose that the response of an organism depends on a number of independentvariables. To investigate this dependency, a common experimental design is to measure the responsewhen changing one variable while keeping all other variable fixed.This experimental design illustrates the idea behind partial derivatives.Suppose we want toknow how the functionf(x, y) changes whenxandychange. Instead of changing both variablessimultaneously, we might get an idea of howf(x, y) depends onxandywhen we change one variablewhile keeping the other variable fixed.To illustrate this we look atExample 1.f(x, y) =x2yWe want to know howf(x, y) changes if we change one variable, sayx, and keep the other variable,in this case,y, fixed. We fixedy=y0, then the change offwith respect toxis simply the derivativeoffwith respect toxwheny=y0. That is,ddxf(x, y0) =ddxx2y0= 2xy0.Such a derivative is called a partial derivative.Date: 2010-03-22.1

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